Integrand size = 442, antiderivative size = 29 \[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\frac {(1+\log (2+x))^2}{x+\left (5+e^{\frac {1}{5+e^2-x}}\right ) x} \]
Time = 0.44 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\frac {e^{\frac {1}{-5-e^2+x}} (1+\log (2+x))^2}{x+6 e^{\frac {1}{-5-e^2+x}} x} \]
Integrate[(-300 + 270*x - 72*x^2 + 6*x^3 + E^4*(-12 + 6*x) + E^2*(-120 + 8 4*x - 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50 + E^4*(-2 + x) + 43*x - 13*x^2 + x^3 + E^2*(-20 + 14*x - 2*x^2)) + (-600 - 24*E^4 + 240*x - 24*x^2 + E^2*( -240 + 48*x) + E^(5 + E^2 - x)^(-1)*(-100 - 4*E^4 + 36*x - 6*x^2 + E^2*(-4 0 + 8*x)))*Log[2 + x] + (-300 + E^4*(-12 - 6*x) - 30*x + 48*x^2 - 6*x^3 + E^2*(-120 - 36*x + 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50 + E^4*(-2 - x) - 7* x + 7*x^2 - x^3 + E^2*(-20 - 6*x + 2*x^2)))*Log[2 + x]^2)/(1800*x^2 + 180* x^3 - 288*x^4 + 36*x^5 + E^4*(72*x^2 + 36*x^3) + E^2*(720*x^2 + 216*x^3 - 72*x^4) + E^(5 + E^2 - x)^(-1)*(600*x^2 + 60*x^3 - 96*x^4 + 12*x^5 + E^4*( 24*x^2 + 12*x^3) + E^2*(240*x^2 + 72*x^3 - 24*x^4)) + E^(2/(5 + E^2 - x))* (50*x^2 + 5*x^3 - 8*x^4 + x^5 + E^4*(2*x^2 + x^3) + E^2*(20*x^2 + 6*x^3 - 2*x^4))),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {6 x^3-72 x^2+e^2 \left (-12 x^2+84 x-120\right )+\left (-24 x^2+e^{\frac {1}{-x+e^2+5}} \left (-6 x^2+36 x+e^2 (8 x-40)-4 e^4-100\right )+240 x+e^2 (48 x-240)-24 e^4-600\right ) \log (x+2)+e^{\frac {1}{-x+e^2+5}} \left (x^3-13 x^2+e^2 \left (-2 x^2+14 x-20\right )+43 x+e^4 (x-2)-50\right )+\left (-6 x^3+48 x^2+e^2 \left (12 x^2-36 x-120\right )+e^{\frac {1}{-x+e^2+5}} \left (-x^3+7 x^2+e^2 \left (2 x^2-6 x-20\right )-7 x+e^4 (-x-2)-50\right )-30 x+e^4 (-6 x-12)-300\right ) \log ^2(x+2)+270 x+e^4 (6 x-12)-300}{36 x^5-288 x^4+180 x^3+1800 x^2+e^4 \left (36 x^3+72 x^2\right )+e^2 \left (-72 x^4+216 x^3+720 x^2\right )+e^{\frac {1}{-x+e^2+5}} \left (12 x^5-96 x^4+60 x^3+600 x^2+e^4 \left (12 x^3+24 x^2\right )+e^2 \left (-24 x^4+72 x^3+240 x^2\right )\right )+e^{\frac {2}{-x+e^2+5}} \left (x^5-8 x^4+5 x^3+50 x^2+e^4 \left (x^3+2 x^2\right )+e^2 \left (-2 x^4+6 x^3+20 x^2\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(\log (x+2)+1) \left (-2 e^{\frac {1}{-x+e^2+5}+2} \left (x^2-7 x+10\right )-12 e^2 \left (x^2-7 x+10\right )-(x+2) \left (e^{\frac {1}{-x+e^2+5}} \left (x^2-9 x+25\right )+6 (x-5)^2-2 e^{\frac {1}{-x+e^2+5}+2} (x-5)-12 e^2 (x-5)+e^{\frac {1}{-x+e^2+5}+4}+6 e^4\right ) \log (x+2)+e^{\frac {1}{-x+e^2+5}} \left (x^3-13 x^2+43 x-50\right )+6 (x-2) (x-5)^2+e^{\frac {1}{-x+e^2+5}+4} (x-2)+6 e^4 (x-2)\right )}{\left (e^{\frac {1}{-x+e^2+5}}+6\right )^2 \left (-x+e^2+5\right )^2 x^2 (x+2)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^3+x^3 (-\log (x+2))-13 \left (1+\frac {2 e^2}{13}\right ) x^2+7 \left (1+\frac {2 e^2}{7}\right ) x^2 \log (x+2)+43 \left (1+\frac {1}{43} e^2 \left (14+e^2\right )\right ) x-7 \left (1+\frac {1}{7} e^2 \left (6+e^2\right )\right ) x \log (x+2)-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right ) \log (x+2)-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right )\right ) (\log (x+2)+1)}{\left (e^{\frac {1}{-x+e^2+5}}+6\right ) \left (-x+e^2+5\right )^2 x^2 (x+2)}+\frac {6 (\log (x+2)+1)^2}{\left (e^{\frac {1}{-x+e^2+5}}+6\right )^2 \left (-x+e^2+5\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {\left (x^3+x^3 (-\log (x+2))-13 \left (1+\frac {2 e^2}{13}\right ) x^2+7 \left (1+\frac {2 e^2}{7}\right ) x^2 \log (x+2)+43 \left (1+\frac {1}{43} e^2 \left (14+e^2\right )\right ) x-7 \left (1+\frac {1}{7} e^2 \left (6+e^2\right )\right ) x \log (x+2)-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right ) \log (x+2)-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right )\right ) (\log (x+2)+1)}{\left (e^{\frac {1}{-x+e^2+5}}+6\right ) \left (-x+e^2+5\right )^2 x^2 (x+2)}+\frac {6 (\log (x+2)+1)^2}{\left (e^{\frac {1}{-x+e^2+5}}+6\right )^2 \left (-x+e^2+5\right )^2 x}\right )dx\) |
Int[(-300 + 270*x - 72*x^2 + 6*x^3 + E^4*(-12 + 6*x) + E^2*(-120 + 84*x - 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50 + E^4*(-2 + x) + 43*x - 13*x^2 + x^3 + E^2*(-20 + 14*x - 2*x^2)) + (-600 - 24*E^4 + 240*x - 24*x^2 + E^2*(-240 + 48*x) + E^(5 + E^2 - x)^(-1)*(-100 - 4*E^4 + 36*x - 6*x^2 + E^2*(-40 + 8* x)))*Log[2 + x] + (-300 + E^4*(-12 - 6*x) - 30*x + 48*x^2 - 6*x^3 + E^2*(- 120 - 36*x + 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50 + E^4*(-2 - x) - 7*x + 7* x^2 - x^3 + E^2*(-20 - 6*x + 2*x^2)))*Log[2 + x]^2)/(1800*x^2 + 180*x^3 - 288*x^4 + 36*x^5 + E^4*(72*x^2 + 36*x^3) + E^2*(720*x^2 + 216*x^3 - 72*x^4 ) + E^(5 + E^2 - x)^(-1)*(600*x^2 + 60*x^3 - 96*x^4 + 12*x^5 + E^4*(24*x^2 + 12*x^3) + E^2*(240*x^2 + 72*x^3 - 24*x^4)) + E^(2/(5 + E^2 - x))*(50*x^ 2 + 5*x^3 - 8*x^4 + x^5 + E^4*(2*x^2 + x^3) + E^2*(20*x^2 + 6*x^3 - 2*x^4) )),x]
3.7.47.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 9.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14
| method | result | size |
| parallelrisch | \(\frac {\ln \left (2+x \right )^{2}+2 \ln \left (2+x \right )+1}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}\) | \(33\) |
| risch | \(\frac {\ln \left (2+x \right )^{2}}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}+\frac {2 \ln \left (2+x \right )}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}+\frac {1}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}\) | \(67\) |
int(((((-2-x)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1/(exp( 2)+5-x))+(-6*x-12)*exp(2)^2+(12*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30*x-300 )*ln(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(exp(2)+5- x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*ln(2+x)+((-2+x)*exp(2) ^2+(-2*x^2+14*x-20)*exp(2)+x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x))+(6*x-12 )*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/(((x^3+2*x^2) *exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp(1/(exp( 2)+5-x))^2+((12*x^3+24*x^2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp(2)+12*x^ 5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*exp(2)^2+(-72 *x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x,method=_RE TURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\frac {\log \left (x + 2\right )^{2} + 2 \, \log \left (x + 2\right ) + 1}{x e^{\left (-\frac {1}{x - e^{2} - 5}\right )} + 6 \, x} \]
integrate(((((-2-x)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1 /(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(12*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30 *x-300)*log(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(ex p(2)+5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*log(2+x)+((-2+x )*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x)) +(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/(((x^ 3+2*x^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp (1/(exp(2)+5-x))^2+((12*x^3+24*x^2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp( 2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*exp(2 )^2+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x, a lgorithm=\
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\frac {\log {\left (x + 2 \right )}^{2} + 2 \log {\left (x + 2 \right )} + 1}{x e^{\frac {1}{- x + 5 + e^{2}}} + 6 x} \]
integrate(((((-2-x)*exp(2)**2+(2*x**2-6*x-20)*exp(2)-x**3+7*x**2-7*x-50)*e xp(1/(exp(2)+5-x))+(-6*x-12)*exp(2)**2+(12*x**2-36*x-120)*exp(2)-6*x**3+48 *x**2-30*x-300)*ln(2+x)**2+((-4*exp(2)**2+(8*x-40)*exp(2)-6*x**2+36*x-100) *exp(1/(exp(2)+5-x))-24*exp(2)**2+(48*x-240)*exp(2)-24*x**2+240*x-600)*ln( 2+x)+((-2+x)*exp(2)**2+(-2*x**2+14*x-20)*exp(2)+x**3-13*x**2+43*x-50)*exp( 1/(exp(2)+5-x))+(6*x-12)*exp(2)**2+(-12*x**2+84*x-120)*exp(2)+6*x**3-72*x* *2+270*x-300)/(((x**3+2*x**2)*exp(2)**2+(-2*x**4+6*x**3+20*x**2)*exp(2)+x* *5-8*x**4+5*x**3+50*x**2)*exp(1/(exp(2)+5-x))**2+((12*x**3+24*x**2)*exp(2) **2+(-24*x**4+72*x**3+240*x**2)*exp(2)+12*x**5-96*x**4+60*x**3+600*x**2)*e xp(1/(exp(2)+5-x))+(36*x**3+72*x**2)*exp(2)**2+(-72*x**4+216*x**3+720*x**2 )*exp(2)+36*x**5-288*x**4+180*x**3+1800*x**2),x)
Time = 0.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\frac {{\left (\log \left (x + 2\right )^{2} + 2 \, \log \left (x + 2\right ) + 1\right )} e^{\left (\frac {1}{x - e^{2} - 5}\right )}}{6 \, x e^{\left (\frac {1}{x - e^{2} - 5}\right )} + x} \]
integrate(((((-2-x)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1 /(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(12*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30 *x-300)*log(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(ex p(2)+5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*log(2+x)+((-2+x )*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x)) +(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/(((x^ 3+2*x^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp (1/(exp(2)+5-x))^2+((12*x^3+24*x^2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp( 2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*exp(2 )^2+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x, a lgorithm=\
\[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\int { \frac {6 \, x^{3} - {\left (6 \, x^{3} - 48 \, x^{2} + 6 \, {\left (x + 2\right )} e^{4} - 12 \, {\left (x^{2} - 3 \, x - 10\right )} e^{2} + {\left (x^{3} - 7 \, x^{2} + {\left (x + 2\right )} e^{4} - 2 \, {\left (x^{2} - 3 \, x - 10\right )} e^{2} + 7 \, x + 50\right )} e^{\left (-\frac {1}{x - e^{2} - 5}\right )} + 30 \, x + 300\right )} \log \left (x + 2\right )^{2} - 72 \, x^{2} + 6 \, {\left (x - 2\right )} e^{4} - 12 \, {\left (x^{2} - 7 \, x + 10\right )} e^{2} + {\left (x^{3} - 13 \, x^{2} + {\left (x - 2\right )} e^{4} - 2 \, {\left (x^{2} - 7 \, x + 10\right )} e^{2} + 43 \, x - 50\right )} e^{\left (-\frac {1}{x - e^{2} - 5}\right )} - 2 \, {\left (12 \, x^{2} - 24 \, {\left (x - 5\right )} e^{2} + {\left (3 \, x^{2} - 4 \, {\left (x - 5\right )} e^{2} - 18 \, x + 2 \, e^{4} + 50\right )} e^{\left (-\frac {1}{x - e^{2} - 5}\right )} - 120 \, x + 12 \, e^{4} + 300\right )} \log \left (x + 2\right ) + 270 \, x - 300}{36 \, x^{5} - 288 \, x^{4} + 180 \, x^{3} + 1800 \, x^{2} + 36 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{4} - 72 \, {\left (x^{4} - 3 \, x^{3} - 10 \, x^{2}\right )} e^{2} + 12 \, {\left (x^{5} - 8 \, x^{4} + 5 \, x^{3} + 50 \, x^{2} + {\left (x^{3} + 2 \, x^{2}\right )} e^{4} - 2 \, {\left (x^{4} - 3 \, x^{3} - 10 \, x^{2}\right )} e^{2}\right )} e^{\left (-\frac {1}{x - e^{2} - 5}\right )} + {\left (x^{5} - 8 \, x^{4} + 5 \, x^{3} + 50 \, x^{2} + {\left (x^{3} + 2 \, x^{2}\right )} e^{4} - 2 \, {\left (x^{4} - 3 \, x^{3} - 10 \, x^{2}\right )} e^{2}\right )} e^{\left (-\frac {2}{x - e^{2} - 5}\right )}} \,d x } \]
integrate(((((-2-x)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1 /(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(12*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30 *x-300)*log(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(ex p(2)+5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*log(2+x)+((-2+x )*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x)) +(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/(((x^ 3+2*x^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp (1/(exp(2)+5-x))^2+((12*x^3+24*x^2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp( 2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*exp(2 )^2+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x, a lgorithm=\
integrate((6*x^3 - (6*x^3 - 48*x^2 + 6*(x + 2)*e^4 - 12*(x^2 - 3*x - 10)*e ^2 + (x^3 - 7*x^2 + (x + 2)*e^4 - 2*(x^2 - 3*x - 10)*e^2 + 7*x + 50)*e^(-1 /(x - e^2 - 5)) + 30*x + 300)*log(x + 2)^2 - 72*x^2 + 6*(x - 2)*e^4 - 12*( x^2 - 7*x + 10)*e^2 + (x^3 - 13*x^2 + (x - 2)*e^4 - 2*(x^2 - 7*x + 10)*e^2 + 43*x - 50)*e^(-1/(x - e^2 - 5)) - 2*(12*x^2 - 24*(x - 5)*e^2 + (3*x^2 - 4*(x - 5)*e^2 - 18*x + 2*e^4 + 50)*e^(-1/(x - e^2 - 5)) - 120*x + 12*e^4 + 300)*log(x + 2) + 270*x - 300)/(36*x^5 - 288*x^4 + 180*x^3 + 1800*x^2 + 36*(x^3 + 2*x^2)*e^4 - 72*(x^4 - 3*x^3 - 10*x^2)*e^2 + 12*(x^5 - 8*x^4 + 5 *x^3 + 50*x^2 + (x^3 + 2*x^2)*e^4 - 2*(x^4 - 3*x^3 - 10*x^2)*e^2)*e^(-1/(x - e^2 - 5)) + (x^5 - 8*x^4 + 5*x^3 + 50*x^2 + (x^3 + 2*x^2)*e^4 - 2*(x^4 - 3*x^3 - 10*x^2)*e^2)*e^(-2/(x - e^2 - 5))), x)
Time = 9.66 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx=\frac {x\,\left (2\,{\ln \left (x+2\right )}^2+4\,\ln \left (x+2\right )+2\right )+x^2\,\left ({\ln \left (x+2\right )}^2+2\,\ln \left (x+2\right )+1\right )}{x^2\,\left ({\mathrm {e}}^{\frac {1}{{\mathrm {e}}^2-x+5}}+6\right )\,\left (x+2\right )} \]
int(-(exp(2)*(12*x^2 - 84*x + 120) - 270*x + log(x + 2)^2*(30*x + exp(2)*( 36*x - 12*x^2 + 120) + exp(1/(exp(2) - x + 5))*(7*x + exp(2)*(6*x - 2*x^2 + 20) + exp(4)*(x + 2) - 7*x^2 + x^3 + 50) - 48*x^2 + 6*x^3 + exp(4)*(6*x + 12) + 300) + log(x + 2)*(24*exp(4) - 240*x + 24*x^2 + exp(1/(exp(2) - x + 5))*(4*exp(4) - 36*x + 6*x^2 - exp(2)*(8*x - 40) + 100) - exp(2)*(48*x - 240) + 600) - exp(1/(exp(2) - x + 5))*(43*x - exp(2)*(2*x^2 - 14*x + 20) + exp(4)*(x - 2) - 13*x^2 + x^3 - 50) + 72*x^2 - 6*x^3 - exp(4)*(6*x - 12) + 300)/(exp(1/(exp(2) - x + 5))*(exp(4)*(24*x^2 + 12*x^3) + exp(2)*(240*x ^2 + 72*x^3 - 24*x^4) + 600*x^2 + 60*x^3 - 96*x^4 + 12*x^5) + exp(4)*(72*x ^2 + 36*x^3) + exp(2)*(720*x^2 + 216*x^3 - 72*x^4) + 1800*x^2 + 180*x^3 - 288*x^4 + 36*x^5 + exp(2/(exp(2) - x + 5))*(exp(4)*(2*x^2 + x^3) + exp(2)* (20*x^2 + 6*x^3 - 2*x^4) + 50*x^2 + 5*x^3 - 8*x^4 + x^5)),x)